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A325923
Number of Motzkin meanders of length n with an odd number of humps and an even number of peaks.
6
0, 0, 0, 1, 5, 18, 56, 163, 459, 1286, 3640, 10479, 30659, 90738, 270092, 804833, 2393929, 7098790, 20984188, 61872587, 182130495, 535698422, 1575478728, 4635125097, 13645054833, 40196623234, 118493318904, 349506908369, 1031426887149
OFFSET
0,5
COMMENTS
A Motzkin meander is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), and never goes below the x-axis.
A peak is an occurrence of the pattern UD.
A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).
FORMULA
G.f.: ( (-3*t^2+4*t+sqrt(-3*t^4+4*t^3+2*t^2-4*t+1)-1)/(3*t^2-4*t+1) + (2*t^3-5*t^2+4*t+sqrt(4*t^6-12*t^5+13*t^4-8*t^3+6*t^2-4*t+1)-1)/(-2*t^3+5*t^2-4*t+1) - (-5*t^2+4*t+sqrt(5*t^4-4*t^3+6*t^2-4*t+1)-1)/(5*t^2-4*t+1) - (-2*t^3-3*t^2+4*t+sqrt(4*t^6+4*t^5-11*t^4+8*t^3+2*t^2-4*t+1)-1)/(2*t^3+3*t^2-4*t+1) ) / (8*t).
a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2019
EXAMPLE
For n = 4 the a(4) = 5 paths are UHDU, UHDH, UUHD, HUHD, UHHD: in all these paths, 0 peaks, 1 hump.
For n=0..6 we have only paths with 0 peaks and 1 hump.
For n=7, we have a(7)=163. Among them, 160 paths with 0 peaks and 1 hump, and 3 walks with 2 peaks and 3 humps: UDUDUHD, UDUHDUD, UHDUDUD.
MAPLE
b:= proc(x, y, t, p, h) option remember; `if`(x=0, `if`(p+1=h, 1, 0),
`if`(y>0, b(x-1, y-1, 0, irem(p+`if`(t=1, 1, 0), 2), irem(h+
`if`(t=2, 1, 0), 2)), 0)+b(x-1, y, `if`(t>0, 2, 0), p, h)+
b(x-1, y+1, 1, p, h))
end:
a:= n-> b(n, 0$4):
seq(a(n), n=0..35); # Alois P. Heinz, Jul 04 2019
MATHEMATICA
CoefficientList[Series[((-1 + 4*x - 3*x^2 + Sqrt[(-(-1 + x)^2)*(-1 + 2*x + 3*x^2)])/ (1 - 4*x + 3*x^2) - (-1 + 4*x - 5*x^2 + 2*x^3 + Sqrt[(-1 + x)^3*(-1 + x + 4*x^3)])/ ((-1 + x)^2*(-1 + 2*x)) + (1 - 4*x + 5*x^2 - Sqrt[1 - 4*x + 6*x^2 - 4*x^3 + 5*x^4])/(1 - 4*x + 5*x^2) + (1 - 4*x + 3*x^2 + 2*x^3 - Sqrt[1 - 4*x + 2*x^2 + 8*x^3 - 11*x^4 + 4*x^5 + 4*x^6])/(1 - 4*x + 3*x^2 + 2*x^3)) / (8*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 09 2019 *)
CROSSREFS
Cf. A325921.
Sequence in context: A325919 A317849 A307572 * A335720 A093374 A258109
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Jul 04 2019
STATUS
approved